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I need to verify that the solution for vanishing Weyl tensor is conformally flat metric $g_{\mu\nu} = e^{2\varphi}\eta_{\mu\nu}$. The most convenient way to show this is to prove that Weyl tensor is invariant under conformal transformation of the metric. How to prove this fast?

I have the idea to build 4-rank tensor which include terms with curvature tensor, Ricci tensor and scalar curvature and then use the requirement on invariance under infinitesimal conformal transformations. If I can show that it is Weyl tensor, I can also prove the statement. But do some alternatives exist?

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Wikipedia has a page which lists how different objects behave under conformal transformations. –  user23660 Nov 21 '13 at 17:16
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You may work with infinitesimal transformations, like in this paper –  Trimok Nov 21 '13 at 18:37

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